Question

A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. If the fencing costs $10 per linear foot to install and the farmer is not willing to spend more than $5000, find the dimensions for the plot that would enclose the most area. (width, length

Answer 1

Refer to the diagram shown below.

x =  the length of the rectangular area
y =  the width of the rectangular area.

The cost of the fence on the western side is shared with a neighbor, so it is $5 per foot. Otherwise, it is $10 per foot. The northern length is not fenced.

Because the cost is not to exceed $5000, therefore
5y + 10x + 10y = 5000
10x + 15y = 5000
x + 1.5y = 500                 (1)

The fenced area is
A = x*y                               (2)

Substitute (1) into (2).
A = (500 - 1.5y)*y = 500y - 1.5y²

To maximize A, A'(y) = 0.
That is,
A'(y) = 500 - 3y = 0
y = 500/3 = 166.67 ft
x = 500 - 1.5y = 250ft

Answer:
The dimensions of the largest area are
Width  = 166.67 ft
Length = 250 ft.

Answer 2

width : 166.67 ft, length : 250 ft

Further explanation

Quadratic function is a function that has the term x²

The quadratic function forms a parabolic curve

The general formula is

f (x) = ax² + bx + c

where a, b, and c are real numbers and a ≠ 0.

The parabolic curve can be opened up or down determined from the value of a. If a is positive, the parabolic curve opens up and has a minimum value. If a is negative, the parabolic curve opens down and has a maximum value

So the maximum is if a <0 and the minimum if a> 0.

The formula for finding the coordinates of the maximum and minimum points of the quadratic function is the same.

The maximum / minimum point of the quadratic function is

$\rm - \dfrac {b} {2a}, - \dfrac {D} {4a}$

Where

D = b²-4ac

Total length of fence :

x + 2y

Because the west side of the plot is shared with a neighbor, the equation becomes:

x + 1.5y

the fencing costs $ 10 per linear foot, and the farmer is not willing to spend more than $ 5000, then the equation becomes:

10x + 15y = 5000

x + 1.5y = 500 ---> x = 500-1.5y

The area of ​​the square :

A = x.y

We input the value x

A = (500-1.5y) y

A = 500y-1.5y²

The maximum area of the quadratic function is (a <0 means the maximum value)

$\rm- \dfrac {D} {4a}$

From the Area equation :

a = -1.5, b = 500, c = 0

D = b²-4ac

D = 500²-4.-1.5.0

D = 500²

D = 250,000

 Then the maximum area:

$\rm A = - \dfrac {D} {4a} \\\\ A = - \dfrac {250,000} {4 \times -1.5} \\\\ A = \boxed {41,666.6 \: ft}}$

• value of x

$\rm x=\dfrac{-b}{2a}\\\\x=-\dfrac{500}{2\times -1.5}\\\\x=166.67$

• value of y

y = A: x

y = 41,666.6: 166.67

y = 250 ft

Learn more  

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